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\title{Population and Stock Hypotheses for North Atlantic Bluefin Tuna}
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\author{Haritz Arrizabalaga  \footnote{AZTI-Tecnalia, Herrera Kaia Portualdea, 20110, Pasaia, Spain; ~gmerino@azti.es; ~Phone: +34 667 174 456 ~Fax: +34 94 657 25 55.},	
        Jean-Marc Fromentin  \footnote{IFREMER - UMR 212 EME (Ecosystèmes Marins Exploités)	 Avenue Jean Monnet - BP 171 - 34203 Sète cedex - France},
        Laurence T. Kell     \footnote{ICCAT Secretariat, C/Coraz\'{o}n de Mar\'{\i}a, 8. 28002 Madrid, Spain; ~Laurie.Kell@iccat.int; ~Phone: +34 914 165 600 ~Fax: +34 914 152 612.},
        Lisa A. Kerr         \footnote{Gulf of Maine Research Institute},
        David H. Secor       \footnote{University of Maryland Center for Environmental Science}
        }
 
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\begin{document}


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\begin{abstract}

\textit{}

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\Keywords{Bluefin, North Atlantic, Mediterranean, Stock, Population, Mixing It}

 
\newpage
\tableofcontents
\newpage

\section[Introduction]{Introduction}

North Atlantic bluefin tuna is currently managed as two separate eastern and western stocks. However, tagging, 
micro-chemistry and genetic studies suggest that the stocks mix and within a stock there are
sub components. Differences are also assumed in life history parameters, e.g. maturity 
and natural mortality, used in their stock assessments.

We review population hypotheses and discuss the importance of the population hypotheses 
for the current and alternative management framework. We then specify  
equations to map the population components into the current and alternative 
management units and discuss how to construct an Operating Model (OM) to be used 
as within a Management Strategy Evaluation (MSE) framework.

The currently assumed stock structure and management units are summarised in figure \ref{map}.

\section{Hypotheses}

\subsection{Populations}

\textit{\bf[Population and life history hypotheses the from BFT-Bio report]}

A variety of population structures have been proposed for North Atlantic bluefin tuna.  The population structures diagrams in figures
 \ref{pop:1},  \ref{pop:2},  \ref{pop:3},  \ref{pop:4} and  \ref{pop:5} were developed during the Bluefin meeting on biological parameters (ref xxx).
These assume that population components are self-reproducing entities (populations and sub-populations) with similar lifetime migration behaviors, 
which do not necessarily depend on reproductive isolation (contingents). The population structures are not intended to be exhaustive: alternative
combinations of sub-populations and contingents could be proposed. The intention is to specify the main potential population structures 
that can be tested with existing data and used to build an OM. 

The five hypotheses are 

\begin{enumerate}
 \item Two population model with no sub-populations
 \item Two population model with contingents
 \item Metapopulation model
 \item Panmictic population with some patchiness
 \item Extended West Atlantic two population model with contingents
\end{enumerate}


\subsection{Life History}

\section{Methodology}

\textit{\bf[Discuss how to construct an OM to test current and alternative management]}

\subsection{Models}

\subsubsection{Variables}

\textit{\bf[List the variables]}

\subsubsection{Equations}

\textit{\bf[Specify the equations]}

\subsection{Implementation}

\subsubsection{Software}

\textit{\bf[How to code in C++ and R]}

\subsubsection{Documentation}

\textit{\bf[Use Oxygen to document all code]}

\subsubsection{Interface}

\textit{\bf[Object Orientating Programming in C++ for developers, and S4 in R for users and MP developers]}

\section[Discussion]{Discussion and Conclusions}

\include{glossary.tex}
\printglossaries

\textit{\bf[A list of all terminology]}


\newpage\clearpage
\bibliography{refs} 
\bibliographystyle{abbrvnat} 

\textit{\bf[A Full literature review]}

\newpage\clearpage\section{Tables}

\begin{tabular}{lp{10cm}}
\toprule
\toprule

% Population dynamics
\textbf{Population dynamics} &
 \begin{equation} N_{a+1, y+1} = N_{a,y} e^{-Z_{a,y}} \end{equation} \\
%
 &
 \begin{equation} N_{p,y} = N_{p-1, y-1} e^{-Z_{p-1, y-1}} + N_{p,y} e ^{-Z_{p, y-1}} \end{equation} \\
%
 &  \begin{equation} N_{r,y} = f(B_{y-r}) \end{equation} \\
\midrule

% Mortality rates
\textbf{Mortality rates} & \begin{equation} Z_{a,y} = F_{a,y} + D_{a,y} + M_{a,y} \end{equation} \\
%
 & \begin{subequations} 
\begin{equation} F_{a,y} = \sum_{i=1}^f P_{i,a,y} S_{i,a,y} E_{i,y} \end{equation}
\begin{equation} D_{a,y} = \sum_{i=1}^f \left(1- P_{i,a,y}\right) S_{i,a,y} E_{i,y} \end{equation}
\end{subequations}\\
\midrule

% Catch equation
\textbf{Catch equation} & \begin{equation} C_{f,a,y} = N_{a,y} \frac{F_{f,a,y}}{Z_{f,a,y}} \left(1 - e^{-Z_{a,y}} \right) \end{equation} \\
\midrule

%
\multicolumn{2}{l}{\textbf{Stock recruitment relationships}} \\
\addlinespace
% BH
Beverton \& Holt & \begin{equation} N_{r,y} = \frac{B_{y-r}}{\alpha B_{y-r} + \beta} \end{equation} \\
% BB
\bottomrule

\multicolumn{2}{l}{\textbf{Growth and maturity}} \\
% BH
von Bertalanffy & \begin{equation} N_{r,y} = \frac{B_{y-r}}{\alpha B_{y-r} + \beta} \end{equation} \\
\bottomrule

\end{tabular}



\newpage\clearpage\section{Figures} 

\begin{figure}[!ht]\begin{center} 
\includegraphics{map.pdf}
\end{center}
\caption{\bf{Spatial strata as informed by distribution, life history, fishery, and management of bluefin tuna}}\label{map}\end{figure} 

\begin{figure}[!ht]\begin{center} 
\includegraphics{pop1.png}
\end{center}
\caption{\bf{Two population model with no sub-populations.}}
\label{pop:1}\end{figure} 


\begin{figure}[!ht]\begin{center} 
\includegraphics{pop2.png}
\end{center}
\caption{\bf{Two population model with contingents.}}
\label{pop:2}\end{figure} 


\begin{figure}[!ht]\begin{center} 
\includegraphics{pop3.png}
\end{center}
\caption{\bf{Metapopulation model.}}
\label{pop:3}\end{figure} 


\begin{figure}[!ht]\begin{center} 
\includegraphics{pop4.png}
\end{center}
\caption{\bf{Panmictic population with some patchiness.}}
\label{pop:4}\end{figure} 


\begin{figure}[!ht]\begin{center} 
\includegraphics{pop5.png}
\end{center}
\caption{\bf{Extended West Atlantic two population model with contingents.}}
\label{pop:5}\end{figure} 

 
\end{document}
